Phase Delay and Group Delay

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The phase response of a filter gives the radian phase shift experienced by each
sinusoidal component of the input signal. Sometimes it is more
meaningful to consider phase
delay[21].

Definition. The phase delay of an
LTI filter with phase response is defined by

The phase delay gives the time delay in seconds experienced by each sinusoidal component of
the input signal. For example, in filter , the phase response is which corresponds to a phase delay which is one-half sample.

and it can be seen that the phase delay expresses phase response as
time delay.

In working with phase delay, care must be taken to ensure all
appropriate multiples of have been included in . We defined simply as the complex angle of the frequencyresponse
, and this is not sufficient for obtaining a phase response
which can be converted to true time delay. By discarding multiples
of , as
is done in the definition of complex angle, the phase delay is
modified by multiples of the sinusoidal period. Since LTI filter
analysis is based on sinusoids
without beginning or end, one cannot in principle distinguish
between “true” phase delay and a phase delay with discarded
sinusoidal periods. Nevertheless, it is convenient to define the
filter phase response as a continuous function of frequency
with the property that (for real filters). This specifies a means of
“unwrapping” the phase response to get a consistent phase delay
curve.

Definition. A more commonly encountered representation of
filter phase response is called the group
delay, and it is defined
by

For linear
phase responses, the group delay and the phase delay are
identical, and each may be interpreted as time delay.

For any phase
function, the group delay may be interpreted as the time delay of the amplitude
envelope of a sinusoid at frequency
[21]. The bandwidth of the amplitude envelope in this
interpretation must be restricted to a frequency interval over
which the phase response is approximately linear. While the proof
will not be given here, it should seem reasonable when the process
of amplitude envelope detection is considered. The narrow “bundle”
of frequencies centered at the carrier frequency
is translated to
Hz. At this point, it is evident that the group delay at the
carrier frequency gives the slope of the linear phase of the
translated spectrum. But this is a constant phase delay, and
therefore it has the interpretation of true time delay for the
amplitude envelope.